On divisibility of the class number

In this paper, criteria of divisibility of the class number of the real cyclotomic field of a prime conductor and of a prime degree by primes the order modulo of which is , are given. A corollary of these criteria is the possibility to make a computational proof that a given does not divide for any (conductor) such that both are primes. Note that on the basis of Schinzel's hypothesis there are infinitely many such primes .

     

Factors of generalized Fermat numbers

A search for prime factors of the generalized Fermat numbers has been carried out for all pairs with and GCD. The search limit on the factors, which all have the form , was for and for . Many larger primes of this form have also been tried as factors of . Several thousand new factors were found, which are given in our tables.-For the smaller of the numbers, i.e. for , or, if , for , the cofactors, after removal of the factors found, were subjected to primality tests, and if composite with , searched for larger factors by using the ECM, and in some cases the MPQS, PPMPQS, or SNFS. As a result all numbers with are now completely factored.

     

On the Diophantine equation

If and are positive integers with and , then the equation of the title possesses at most one solution in positive integers and , with the possible exceptions of satisfying , and . The proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometric functions, the theory of linear forms in logarithms and recent computational methods related to lattice-basis reduction. Additionally, we compare and contrast a number of these last mentioned techniques.

     

A conjecture of Erdös on 3-powerful numbers

Erdös conjectured that the Diophantine equation has infinitely many solutions in pairwise coprime 3-powerful integers, i.e., positive integers for which implies . This was recently proved by Nitaj who, however, was unable to verify the further conjecture that this could be done infinitely often with integers , and none of which is a perfect cube. This is now demonstrated.

     

Class number bounds and Catalan's equation

We improve a criterion of Inkeri and show that if there is a solution to Catalan's equation

with and prime numbers greater than 3 and both congruent to 3 , then and form a double Wieferich pair. Further, we refine a result of Schwarz to obtain similar criteria when only one of the exponents is congruent to 3 . Indeed, in light of the results proved here it is reasonable to suppose that if , then and form a double Wieferich pair.

     

Steiner systems

It is proved that there are precisely 4204 pairwise non-isomorphic Steiner systems invariant under the group and which can be constructed using only short orbits.

It is further proved that there are precisely 38717 pairwise non-isomorphic Steiner systems invariant under the group and which can be constructed using only short orbits.

     

Factoring elementary groups of prime cube order into subsets

Let be a prime and let be the -fold direct product of the cyclic group of order . Rédei conjectured if is the direct product of subsets and , each of which contains the identity element of , then either or does not generate all of . The paper verifies Rédei's conjecture for .

     

Computations of class numbers of real quadratic fields

In this paper an unconditional probabilistic algorithm to compute the class number of a real quadratic field is presented, which computes the class number in expected time . The algorithm is a random version of Shanks' algorithm. One of the main steps in algorithms to compute the class number is the approximation of . Previous algorithms with the above running time , obtain an approximation for by assuming an appropriate extension of the Riemann Hypothesis. Our algorithm finds an appoximation for without assuming the Riemann Hypothesis, by using a new technique that we call the `Random Summation Technique'. As a result, we are able to compute the regulator deterministically in expected time . However, our estimate of on the running time of our algorithm to compute the class number is not effective.

     

Power series with restricted coefficients and a root on a given ray

We consider bounds on the smallest possible root with a specified argument of a power series with coefficients in the interval . We describe the form that the extremal power series must take and hence give an algorithm for computing the optimal root when is rational. When we show that the smallest disc containing two roots has radius coinciding with the smallest double real root possible for such a series. It is clear from our computations that the behaviour is more complicated for smaller . We give a similar procedure for computing the smallest circle with a real root and a pair of conjugate roots of a given argument. We conclude by briefly discussing variants of the beta-numbers (where the defining integer sequence is generated by taking the nearest integer rather than the integer part). We show that the conjugates, , of these pseudo-beta-numbers either lie inside the unit circle or their reciprocals must be roots of power series; in particular we obtain the sharp inequality .

     

Classification of integral lattices with large class number

A detailed exposition of Kneser's neighbour method for quadratic lattices over totally real number fields, and of the sub-procedures needed for its implementation, is given. Using an actual computer program which automatically generates representatives for all isomorphism classes in one genus of rational lattices, various results about genera of -elementary lattices, for small prime level are obtained. For instance, the class number of -dimensional -elementary even lattices of determinant is ; no extremal lattice in the sense of Quebbemann exists. The implementation incorporates as essential parts previous programs of W. Plesken and B. Souvignier.

     

Finding finite -sequences faster

A -sequence is a sequence of positive integers such that the sums , , are different. When is a power of a prime and is a primitive element in then there are -sequences of size with , which were discovered by R. C. Bose and S. Chowla.

In Theorem 2.1 I will give a faster alternative to the definition. In Theorem 2.2 I will prove that multiplying a sequence by integers relatively prime to the modulus is equivalent to varying . Theorem 3.1 is my main result. It contains a fast method to find primitive quadratic polynomials over when is an odd prime. For fields of characteristic 2 there is a similar, but different, criterion, which I will consider in ``Primitive quadratics reflected in -sequences', to appear in Portugaliae Mathematica (1999).

     

Minus class groups of the fields of the -th roots of unity

We show that for any prime number the minus class group of the field of the -th roots of unity admits a finite free resolution of length 1 as a module over the ring . Here denotes complex conjugation in . Moreover, for the primes we show that the minus class group is cyclic as a module over this ring. For these primes we also determine the structure of the minus class group.

     

An algorithm for constructing a basis for -spline modules over polynomial rings

Let be a polyhedral complex embedded in the euclidean space and , , denote the set of all -splines on . Then is an -module where is the ring of polynomials in several variables. In this paper we state and prove the existence of an algorithm to write down a free basis for the above -module in terms of obvious linear forms defining common faces of members of . This is done for the case when consists of a finite number of parallelopipeds properly joined amongst themselves along the above linear forms.

     

Remarks on the Schoof-Elkies-Atkin algorithm

Schoof's algorithm computes the number of points on an elliptic curve defined over a finite field . Schoof determines modulo small primes using the characteristic equation of the Frobenius of and polynomials of degree . With the works of Elkies and Atkin, we have just to compute, when is a ``good" prime, an eigenvalue of the Frobenius using polynomials of degree . In this article, we compute the complexity of Müller's algorithm, which is the best known method for determining one eigenvalue and we improve the final step in some cases. Finally, when is ``bad", we describe how to have polynomials of small degree and how to perform computations, in Schoof's algorithm, on -values only.

     

Multilevel Additive Schwarz Method for the Version of the Galerkin Boundary Element Method

We study a multilevel additive Schwarz method for the - version of the Galerkin boundary element method with geometrically graded meshes. Both hypersingular and weakly singular integral equations of the first kind are considered. As it is well known the - version with geometric meshes converges exponentially fast in the energy norm. However, the condition number of the Galerkin matrix in this case blows up exponentially in the number of unknowns . We prove that the condition number of the multilevel additive Schwarz operator behaves like . As a direct consequence of this we also give the results for the -level preconditioner and also for the - version with quasi-uniform meshes. Numerical results supporting our theory are presented.

     

Evaluation of discrete logarithms in a group of

We show that to solve the discrete log problem in a subgroup of order of an elliptic curve over the finite field of characteristic one needs operations in this field.

     

Fast evaluation of multiple zeta sums

We show that the multiple zeta sum:

for positive integers with , can always be written as a finite sum of products of rapidly convergent series. Perhaps surprisingly, one may develop fast summation algorithms of such efficiency that the overall complexity can be brought down essentially to that of one-dimensional summation. In particular, for any dimension one may resolve good digits of in arithmetic operations, with the implied big- constant depending only on the set .

     

Computing

Let denote the Von Mangoldt function and . We describe an elementary method for computing isolated values of . The complexity of the algorithm is time and space. A table of values of for up to is included, and some times of computation are given.

     

Quadrature formulae using zeros of Bessel functions as nodes

A gaussian type quadrature formula, where the nodes are the zeros of Bessel functions of the first kind of order (), was recently proved for entire functions of exponential type. Here we relax the restriction on as well as on the function. Some applications are also given.

     

An algorithm for evaluation of discrete logarithms in some nonprime finite fields

In this paper we propose an algorithm for evaluation of logarithms in the finite fields , where the number has a small primitive factor . The heuristic estimate of the complexity of the algorithm is equal to
, where grows to , and is limited by a polynomial in . The evaluation of logarithms is founded on a new congruence of the kind of D. Coppersmith, , which has a great deal of solutions-pairs of polynomials of small degrees.